Twisted Yangian的表示

表示理论的核心问题之一是构造表示的一组基并给出生成元在基上的作用。在1950年Gelfand和Tsetlin构造了一般线性李代数和正交李代数的有限维不可约表示的基，这些基现在被称为Gelfand-Tsetlin基。随后Gelfand-Tsetlin基被推广到量子群，李超代数，Yangian等。我们将构造twisted Yangian有限维不可约表示的Gelfand-Tsetlin基，并显示地给出生成元在基上的作用。另外一个构造twisted Yangian表示的方法是将典型李代数有限维表示的张量积看做是twisted Yangian的表示，然后考虑张量积的不可约性。我们将给出张量积不可约的充分必要条件。最后我们将有限维表示的结论推广到无限维表示。

One of the central problems of the representation theory is to construct a basis in the representation and find the explicit action of generators on the basis. In 1950, Gelfand and Tsetlin described bases of irreducible finite-dimensional representations over the general linear Lie algebra and orthogonal Lie algebra. The bases are called the Gelfand-Tsetlin bases now. Some steps toward a generalization of the Gelfand-Tsetlin basis have been taken for quantum groups, Lie superalgebras, Yangians and so on. We will construct the Gelfand-Tsetlin bases for finite-dimensional irreducible representations of twisted Yangians and give the explicit action of the generators. A different approach to construct representations of twisted Yangians is to regard the tensor product of finite-dimensional irreducible representations of classical Lie algebras as representations of twisted Yangian and consider the irreducibility of the tensor product. We will give the necessary and sufficient condition for the irreducibility of the tensor products. We will also generalize the result of finite-dimensional representations to infinite-dimensional representations.